Proof and Problem Solving at University Level
Proof and Problem Solving at University Level
复制标题
大学水平的证明和问题解决
DOI：
10.54870/1551-3440.1269
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. Selden
中科院分区：
文献类型：
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作者：
Annie Selden;J. Selden
摘要：本文将关注本科生和研究生的问题解决，因为他们遇到它在试图证明定理，主要是为了满足他们的教授在他们的课程，但也作为他们进行原创性的研究论文和学位论文。我们采取Schoenfeld（1985）的问题观，即一个数学任务是一个问题，如果这个人还不知道解决这个任务的方法。因此，一个给定的任务可能是一个人的问题，谁还不知道解决方法的任务，或者它可能是一个练习的个人谁已经知道一个程序或算法来解决这个任务.关键词：匈牙利，数学教育，数学竞赛，奥林匹克竞赛，国际比较数学教育，问题解决，创造力，数学天才的学生。从非常常规到非常非常规的连续任务虽然什么是问题取决于解决者知道什么，但大多数数学教师可以判断给定班级中大多数学生的困难。因此，我们看到一个给定类的数学任务，例如微积分类，从那些非常常规的问题到那些真正困难的问题（Seiden，Seiden，Hauk，& Mason，2000）。一方面，有一些非常常规的问题，它们模仿教科书或讲座中的样本问题，除了在措辞、符号、系数、常数或函数上的微小变化，这些变化是解决问题的方式所附带的。这类问题通常被称为习题（在解题文献中可能根本不被认为是问题）。微积分教科书中的绝大多数习题都是这种性质的。Lithner（2004）区分了典型微积分教科书练习的三种可能的解决策略：相似性识别（IS），局部合理推理（LPR）和全局合理推理（GPR）。在IS中，人们识别练习的表面特征，并寻找类似的教科书情况-例子，规则，定义，定理。如果不考虑内在的数学性质，人们只是简单地复制那种情况的过程。在LPR中，人们识别出稍微类似于教科书的情况，但是其中一些局部部分可能不同。解决方案策略是尽可能多地从类似的情况中复制，并根据需要修改本地步骤。在GPR中，该策略主要基于分析和考虑练习的内在数学属性，并使用这些，构建解决方案并由合理推理支持。Lithner选择了瑞典使用的教科书[亚当斯的《微积分：一门完整的课程》（第5版），Addison-Wesley]，并对598个单变量微积分练习的解决策略进行了分类。他发现85%的IS，8%的LPR和7%的GPR。此外，他总结道：“在大约70%的练习中，解决方案不仅可以基于搜索类似的情况，而且可以只搜索已解决的例子。“向连续体的中间移动，有适度的常规问题，虽然不完全像样本工作问题，可以通过良好实践的方法来解决，例如，微积分课程中的普通相关速率或变量积分问题的变化。1沿着连续体进一步沿着移动，有适度的非常规问题，这些问题与学生以前见过的问题不太相似，并且需要以稍微新颖的方式组合已知的事实或技能，但是在不需要例如考虑多个子问题或新颖见解方面是“直接的”。这是我们在三项本科生微积分问题解决研究中的非常规测试中使用的问题类型。其中一个问题是：找到a和B的值，使得2x+3 y =a的直线在x=3的点处与f（x）= bx 2的曲线相切。…
Abstract: This paper will be concerned with undergraduate and graduate students' problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld's (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task.Keywords: Hungary, mathematics education, mathematics competition, Olympiads, international comparative mathematics education, problem solving, creativity, mathematically talented students.A Continuum of Tasks from Very Routine to Very Non-routineWhile what is a problem depends on what a solver knows, it is possible for most mathematics teachers to judge what is difficult for most students in a given class. Thus, we see mathematical tasks for a given class, such as a calculus class, on a continuum from those that are very routine to those that are genuinely difficult problems (Seiden, Seiden, Hauk, & Mason, 2000). At one end, there are very routine problems which mimic sample worked problems found in textbooks or lectures, except for minor changes in wording, notation, coefficients, constants, or functions that are incidental to the way the problems are solved. Such problems are often referred to as exercises (and might not be considered to be problems at all in the problem-solving literature).The vast majority of exercises in calculus textbooks are of this nature. Lithner (2004) distinguished three possible solution strategies for typical calculus textbook exercises: identification of similarities (IS), local plausible reasoning (LPR), and global plausible reasoning (GPR). In IS, one identifies surface features of the exercise and looks for a similar textbook situation - an example, a rule, a definition, a theorem. Without consideration of intrinsic mathematical properties, one simply copies the procedure of that situation. In LPR, one identifies a slightly similar textbook situation, but one in which a few local parts may differ. The solution strategy is to copy as much as possible from that similar situation, modifying local steps as needed. In GPR, the strategy is mainly based on analyzing and considering intrinsic mathematical properties of the exercise, and using these, a solution is constructed and supported by plausible reasoning. Lithner selected a textbook used in Sweden [Adams' Calculus: A Complete Course (5th ed.), Addison-Wesley], and worked through and classified solution strategies for 598 single-variable calculus exercises. He found 85% IS, 8% LPR, and 7% GPR. Furthermore, he concluded that "it is possible in about 70% of the exercises to base the solution not only on searching for similar situations, but on searching only the solved examples."Moving toward the middle of the continuum, there are moderately routine problems which, although not exactly like sample worked problems, can be solved by well-practiced methods, for example, ordinary related rates or change of variable integration problems in a calculus course.1 Moving further along the continuum, there are moderately non-routine problems, which are not very similar to problems that students have seen before and require known facts or skills to be combined in a slightly novel way, but are "straightforward" in not requiring, for example, the consideration of multiple sub-problems or novel insights. This is the type of problem we used on the non-routine test in our three studies of undergraduate students' calculus problem solving. One of those problems was: Find values of a and b so that the line 2x+3y=a is tangent to the graph of f (x) = bx2 at the point where x=3. …